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Helicity basis

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In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example). In this basis, the spin is quantized along the axis in the direction of motion of the particle.

Spinors

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The two-component helicity eigenstates satisfy

where
are the Pauli matrices,
is the direction of the fermion momentum,
depending on whether spin is pointing in the same direction as or opposite.

To say more about the state, we will use the generic form of fermion four-momentum:

Then one can say the two helicity eigenstates are

and

These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather:

.

In this situation the helicity eigenstates are for when the particle momentum is

and

then for when momentum is

and

Fermion (spin 1/2) wavefunction

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A fermion 4-component wave function, may be decomposed into states with definite four-momentum:

where
and are the creation and annihilation operators, and
and are the momentum-space Dirac spinors for a fermion and anti-fermion respectively.

Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is

and for an anti-fermion,

Dirac matrices

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To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.

Spin-1 wavefunctions

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The plane wave expansion is

.

For a vector boson with mass m and a four-momentum , the polarization vectors quantized with respect to its momentum direction can be defined as

where
is transverse momentum, and
is the energy of the boson.